As the feature size of semiconductor devices becomes smaller and smaller, the variations in a semiconductor device and VLSI circuit become larger and larger. Variations include systematic variations and random variations. Random variations (uncorrelated variations) affect relative characteristics of closely placed identical devices (often called matching or mismatch), and also affect the relative characteristic of identical devices and circuits placed in several locations within a die (often called across chip variations).
Other circuit and device examples involving random variations include the mismatch among multiple identical NFETs used in a ring oscillator, the mismatch among multiple identical PFETs used in a ring oscillator, and the mismatch among multiple identical fingers in a multi-finger (RF) FET or MOS varactor. From the view point of modeling, systematic variations moves all instances of a device (e.g., all NFET gate oxide thickness) together, and there is no relative difference between any two instances of a devices. This is essentially a one-body (one-instance) statistical modeling problem.
On the other hand, random variations move each instance differently from any other instances (e.g., FET Vt variation caused by random doping fluctuations, the random ACV part of FET channel length variation, mismatch among a group of nearby and identical resistors), and the relative difference between any two instances are the very subject of ACV/mismatch modeling. This is a two-body (two-instance) statistical modeling problem.
For a given device length and width, the known Monte Carlo model for ACV/mismatch is usually straightforward. However, the known Monte Carlo model is not an irreducible representation, i.e., is not the most compact representation. For example, for a two-instance case, the Monte Carlo model is in a three-dimensional space, and there is no one-to-one mapping relation between instance values and their representation in Monte Carlo simulation space. Although it does not affect Monte Carlo simulation results, this does lead to inferior corner models, due to their low joint probability. Also, this extra dimension causes a lot of confusion and uncertainty on which corner is the needed corner, due to the lack of a one-to-one mapping relation. These are explained in details in the following examples of known Monte Carlo statistical model for mismatch/across-chip variations (ACV).